This
research was designed to create computer-based algebra lessons using
spreadsheets about linear functions with emphasis on multiple representations
and to investigate possible effects of instructional uses of multiple representations
on students’ outcomes (attitudes and achievement).This chapter reviews literature relevant to this study and
presents a theoretical framework for the research.

The
chapter is divided into four main sections.The first section presents research concerning the use of multiple
representations in mathematics.It will
briefly discuss research studies dealing with the following topics: (a) need to
use representations in mathematics education, as well as, some of their
strengths and weaknesses; (b) definitions and classifications of
representations; (c) students’ preferences for using representations; (d)
connections among representations; and (e) interpretation of
representations.The second section
describes related learning theories that support the use of multiple
representations. The third section discusses the role of technology and the use
of representations in mathematics.The
fourth section of this chapter contains research studies supporting the teaching
of functions using representations.

Needs to Use Multiple
Representations in Mathematics

The
uses of multiple representations have been strongly connected with the complex
process of learning in mathematics, and more particularly, with the seeking of
the students’ better understanding of important mathematical concepts. Research
done by Hiebert and Carpenter, (1992); Kaput, (1989a); and Skemp, (1987)
illustrates that multiple representations of concepts can be utilized as a help
for students in order to develop deeper, and more flexible understandings
(Porzio, 1994).As cited in Gningue
(2000, p. 43), Kaput (1989a) “thinks that students learn through several modes
of representations”.Dufour-Janvier,
Bednarz, and Belanger (1987) have described important elements about the uses
of representations in mathematics. Dufour-Janvier and colleagues argue that
representations are inherent in mathematics; they are multiple concretizations
of a concept; they could be used to mitigate certain difficulties; and they are
intended to make mathematics more attractive and interesting (pp.110-111).Keller and Hirsch (1998) describe some potential
benefits [italics added] related to the use of representations.Among these benefits are: (a) provide
multiple concretizations of a concept, (b) selective emphasis and de-emphasis
different aspects of complex concepts, and (c) facilitate cognitive linking of
representations (p. 1).Kaput (1992)
points out that the use of multiple representations or notations could be
helpful at the time of present a clear and better picture of a concept or idea.

Complex ideas are seldom adequately represented
using a single notation system… Each notation system reveals more clearly than
its companion some aspect of the idea while hiding some other aspects. The
ability to link different representations helps reveal the different facets of
a complex idea explicitly and dynamically. (p. 542)

De Jong, et
al. (1998) argue that in today’s educational processes, students have been
“confronted with information from different sources (computer programs, books,
the teacher, reality, the classroom, peers, etc.) and in many different
representations that they have to evaluate, make a selection from, and
integrate them into their personal knowledge construction process” (p. 9).About this particular, Poppe (1993) says
that the wide uses of mapping diagrams, graphs, and tables, provide a visual
representation of the relationships between quantities.

The uses of representations in mathematics
have not been a really new trend in educational practices. Porzio (1994)
indicates that mathematics educators have made efforts for the last years, in
order to use more than one representation to introduce mathematical concepts to
students.Janvier, Girardon, and Morand
(1993) point out that educators and researchers have emphasized, through the
years,the roles of different forms of
representation illustrated as: graphs, tables, diagrams, charts and figures.Current reform efforts in various curriculum
projects dealing with calculus instruction at college level, demonstrate that
multiple representations have played a particular role in these processes.As cited in Hart (1991, p. 2), “this
emphasis on multiple representations fits the picture of calculus reform in
which Tucker (1987, p. 16) sees ‘a vista of a more conceptual, intuitive,
numerical, pictorial calculus’ as the calculus of tomorrow”.

De Jong, et al. (1998) stated that
there are three goals that multiple representations serve.First, multiple representations are
recommended to use due to the information that students learn has varied
characteristics.Second, multiple
representations are good resources to induce in the students a particular
quality in their knowledge. De Jong and colleagues say, “both approaches lead
to a concurrent presentation of multiple representations” (p. 39).And third, it is an assumption that the use
of representations in sequence is beneficial for learning.This last goal illustrates the transitional
presentation of representation.Furthermore, these researchers have identified four factors thatmediate the effects of using
representations.

The type of test used is partly responsible for the
effects… The type of domain in the learning environment may also be of
influence… The type of learner using the environment also influences the
effectiveness, and finally, the type of support present in the environment also
plays a role. Most environments simply assume that the co-presence of more than
one representation will prompt the learner to integrate the information. (p.
39)

In addition, De Jong, et al. (1998)
identified three reasons (explicitly or implicitly) about the uses of
representations.The first reason deals
with what to do with the tuning of the domain information and the
representation.The second concerns the
idea that the use of multiple representations will promote a flexible
knowledge.And lastly, the specific order
that representations are introduced into learning will facilitate it.

The first reason for using multiple representations
is that specific information can be conveyed in a specific representation, and
that for a complete set of learning material, containing a variety of
information, a combination of several representations is therefore
necessary.The main issue here is that
of adequacy, which concerns the expressional possibilities of a
representation.A second aspect that
can be involved here is efficiency, which concerns the expressional power of a
representation. Within one level of adequacy, e.g., graphical representations,
some may still be more efficient than others.The second reason for using more than one representation is that
expertise is quite often seen as the possession and coordinated use of multiple
representations of the same domain.In
this theory expertise is viewed as being able to understand the domain
knowledge from multiple perspectives… The third reason for using more than one
representation is based on the assumption that a specified sequence of learning
material is beneficial for the learning process. (pp. 32-33)

Greeno and Hall (1997) call tools
[italics added] the forms of representations in mathematics. They argue that
students can learn to use them “as resources in thinking and communicating” (p.
362).Porzio (1994) citing the research
of Dufour-Janvier and colleagues (1987) says that it is desired that “students
can perceive representations as mathematical tools for solving problems and
helping students in the ‘construction’ of a concept by viewing common
properties and differences between representations of the concept.The research work group headed by
Dufour-Janvier has explored three important categories concerning
representations and have raised a group of questions dealing with each one of these
categories: (a) how these tools have been used in mathematics instruction; (b)
how are the expected outcomes achieved in the current teaching of mathematics;
and (c) how should be the representations to be useful in mathematics.

Dufour-Janvier, et al. (1987) have realized that mathematics teaching,
together with all the elements including in its curricula have submitted
students, of all ages and school levels, to a wide variety of representations.At this point, these researchers propose the
following questions:

What are the motives for using external
representations in mathematics teaching? What are the expected outcomes that
justify such a wide variety of representations? Are these outcomes achieved in
current teaching of mathematics? To what extent is it possible that such
representations are inaccessible to students and even detrimental? Can the
teaching of mathematics be organized in such a way that learning is articulated
with the representations children develop themselves? (p. 109)

These recognized scholars have, also
looked at the outcomes of the uses of representations in the learning of
mathematics.Dufour-Janvier and
colleagues (1987) present some expectation concerning the uses of representations.They expect first, that in particular
mathematics problem situations, students could be able to reject one
representation in order to choose another one, knowing the reasons because they
are doing this selection.Second, it is
expected that students could pass from one representation to another, knowing
the possibilities, limits and effectiveness of each one. Third, students should
be able to select the appropriate representation taking into consideration the
task.Finally, through the use of
multiple representations, students will be able to “grasp the common properties
of these diverse materials and will succeed in constructing the concept” (p.
111).

Another group of questions that
Janvier and colleagues (1987) focused in their research were the following:

1.Does
the students “select” a representation? Among several representations presented
to them, do they know which one to retain, which is the most appropriate to
accomplish the task?

2.Do
the students see the same task in each of the representations given?

3.Are
the students convinced that regardless of the particular representation they
use as an aid to solve a problem they will necessarily arrive at the same
result?

4.How
do students develop the attitude of having recourse to representations in case
they encounter difficulties? (p. 114)

The literature, supported by the extensive research done by Janvier, et
al. (1987) has raised these questions, summarized above, and many others
regarding the usefulness of representations in mathematics.The base of their concern and many of their
inquiries is in the fact that current teaching practices using representations
are not fulfilling their objectives and moreover, their contribution to the
learning process is almost null.“Certain representations lead more to difficulties rather than
functioning as aids to learning” (p. 116).

Greeno and Hall (1997) explored the argument about how representational
forms should be made and used in innovative classroom settings.First, they affirm that representations are
constructed for specific purposes in order to attempt to solve problems and
communicate with others about it.Second, students frequently develop representations with the purpose of
observing patterns and performing mathematical procedures, keeping in mind the
fact that different forms provide different supports.Lastly, students frequently use multiple representations in order
to solve a problem.Some of the
representations used by students are constructed by themselves and they could
differ considerably from the representations taught in the curriculum.

Some Weaknesses of Using Representations in Mathematics

Lines of research studies describe some weaknesses or disadvantages of
the uses of representations in mathematics teaching and learning.Poppe (1993) exploring the effects of
differing technological approaches to calculus on students’ use and
understanding of multiple representations when solving problems, found that
although students realized that tables, graphs, and mapping diagrams were
helpful, they did not use them in order to solve unfamiliar mathematical
problems unless suggested to do so.Dufour-Janvier and colleagues (1987) investigating the accessibility of
representations concluded that the use of representations is sometimes abstract
to students, and this could provoke a lack of meaning to them.Also, they affirm that the inappropriate
context use of representations, as well as the prematurely of their use,
resulting in negative consequences to students.“The use of such nonaccessible representations encourages a play
on symbols, puts the emphasis on the syntactical manipulations of symbols
without reference to the meaning.The
signified is absent!Mathematics is
reduced to a formal language” (p. 11).

Van Someren, et al. (1998) conducting research in multiple representations
in teaching, affirm that the use of combined representations in mathematics
“creates new problems for the learner” (p. 4).They go beyond by saying that multiple representations are not a good
thing per se [italics added].These
researchers claim that when information is presented to students in varied
forms, it is particularly important to also teach the relations or connections
between representations since, if students are left alone to construct them
themselves, it will be difficult.Finally,
Van Someren and colleagues call for a need for a closer analysis between their
semantic relations and performance characteristics, in order to appropriately
use multiple representations in problem solving.

Definitions of Representations

Until this point, the research has showed the need to use representations
in mathematics teaching and learning.It is important to look at how the literature has defined
representations.There are few
researchers who have attempted to define representations in mathematics.The only clear definition comes from the
work done by Özgün-Koca (1998) who stated that “multiple representations are
defined as external mathematical embodiments of ideas and concepts to provide
the same information in more than form” (p. 1).Another definitions could not be found.

Classification of Representations

Nevertheless, the literature does show some research studies concerning
the classification of representations.Porzio (1994, p. 3), citing the work done by Dufour-Janvier and colleagues
(1987), classifies representations as external and internal.

Internal representations concern most particularly
mental images corresponding to internal formulations we construct of reality
(we are here in the domain of the signified).External representations refer to all external symbolic organizations
(symbol, schema, diagrams, etc.) that have as their objective to represent
externally a certain mathematical ‘reality’. (p. 109)

Lesh, Post,
and Behr (1987) have said that external representations are the way by which
mathematical ideas could be communicated and they are presented as physical
objects, pictures, spoken language, or written symbols.

The research group headed by Janvier
(1993), a recognized scholar in this field, expanded the idea of classification
of representations.

External representations act as stimuli on the
senses and include charts, tables, graphs, diagrams, models, computer graphics,
and formal symbol systems.They are
often regarded as embodiments of ideas or concepts.The nature of internal representations is more illusive, because
they cannot be directly observed. (p. 81)

They affirm
that important concepts in a representation theory are “to mean” or “to signify”
(p. 81).In this way, Janvier and
colleagues state that external representation, which they call signifier, and
internal representation, called signified, should be linked.

Cuoco (2001) affirms that:

External representations are the representations we
can easily communicate to other people; they are the marks on the paper, the
drawings, the geometry sketches, and the equations.Internal representations are the images we create in our minds
for mathematical objects and processes – these are much harder to describe. (p.
x)

Goldin and Shteingold (2001) expand the discussion on the types of
representation arguing that:

External systems of representation range from the
conventional symbol systems of mathematics (such as base-ten numeration, formal
algebraic notation, the real number line, or Cartesian coordinate
representation) to structured learning environments (for example, those
involving concrete manipulative materials or computer-based micro worlds).Internal systems, in contrast, include
students’ personal symbolization constructs and assignments of meaning to
mathematical notations, as well as their natural language, their visual imagery
and spatial representation, their problem-solving strategies and heuristics,
and (very important) their affect in relation to mathematics. (p. 2)

Janvier and colleagues (1993)
emphasizing the classification of representations introduced the term “iconic”.
They say that external representations could be iconic since “they can more or
less suggest in their arrangement or configuration the internal representation
to which they relate” (p. 82).These
researchers consider the term “symbolic” as equivalent to the word
“noniconic”.They explain that the
symbolism of an external representation depends primarily on the arbitrary
arrangement or the selection of elements, which constitute it.When any other feature has not helped the
interpretation process, it refers to as noniconic representations.Janvier, et al. affirms that the majority of
mathematics representations could be classified as noniconic.

The psychologist Jerome Bruner,
using some guidelines investigated by Piaget, has been considered as one of the
first researchers who implicitly classified representations.Bruner (1964) proposed three modes of representation:
(a) enactive, (b) iconic, and (c) symbolic.Using the modes of representation introduced by Bruner, Mason (1987a) he
has presented the idea that teaching schemes are a spiral movement.As they pass through the spiral, students
will go from using manipulable external representations to gain a meaning of
internal representations to symbolic representations.Mason proposes that one aim should be to help students to
construct internal representations strongly related to external representations
where they feel confident.

As discussed by Janvier, et al. (1993) another line of research regarding
classification of representations comes from the studies done by Bertin (1967)
who used three categories. The first one maps, which includes the
representation that keeps a fair degree of similarity with the special
properties of the objects they represent.The second, which shows the nature of the relations between variables,
is called diagrams.Familiar
mathematical concepts such as data charts, graphs, belong to this
category.Lastly, networks refer to
when representations of this class show the relationships between events,
factors, or individuals (pp. 82-83).

Janvier and colleagues (1993) have
realized that the existence of many representations in mathematics is a cause
of confusion on students.Trying to
relate internal and external representations in mathematics, they propose two
important terms in their discussion: homonymy and synonymy
[italics added].The first phenomenon
in mathematics is found when one representation has two different
meanings.That is, from an external
representation there are two different internal representations.The second term refers to when one mental
object is denoted in many representations: from two different external representations
there is one internal representation.According to their findings, homonymy, as well as synonymy cannot be
avoided in mathematics.“They belong to
it per se” (p. 88).

Students’ Preferences for Representations

It
is frequently observed that students in the classroom show certain preferences
for one particular external representation.The literature contains important research studies concerning
preferences exhibited by students in order to select a representation.Hart (1991), who developed extensive
research concerning representations, explored their management.She studied students’ preferred
representations and how they vary the choice of representation depending on the
problem.Hart found that there are
factors that influence students’ choice of representation.Her findings are summarized in the following
points:

1.Students
confident in their symbolic manipulation skills tend to use alternate
representations only when unsuccessful at finding an answer symbolically.

2.Students
make a choice of representations depending on the complexity of the symbolic
information provided.

3.Some
students do not use a certain representation because they do not recognize that
it’s a viable choice.

4.Students
lack confidence in using certain representations.

5.Students
who do not have access to a graphing calculator do not typically choose to use
the graphical representation.

Hart’s
findings indicate that the representation used by students to solve problems is
strongly influenced by their previous experiences.

Research by Yerushalmy (1997)
revealed, “normally, symbolic (formula or equation) representation is the more
convenient representation for modeling situations with two independent
variables.However, the priorities for
students who have not yet learned to manipulate symbols but have experienced
modeling through various other representations could be different” (p. 432).

Keller and Hirsch (1998) identified
two types of research on students’ preferences for representation.The first line of research deals
particularly with the attempt to determine students’ preferences by the
representation used to perform tasks.LaLomia, Coovert and Salas (1988) conducted research regarding which of
two types of representation – tables and graphs – students used most often to
solve tasks.Their findings show that
students preferred tables when they had to locate particular numbers.On the other hand, students only slightly
used graphs with interpolation and forecasting tasks.The second line of research dealing with preferences in
representations concerns learning theories or cognitive styles.About this second line of research on
representations’ preferences, Turner and Wheatley (1980) explored the
preferences of students in an elementary calculus course emphasizing two
representations: graphical and linguistic.They found that students exhibited strong preferences for each
form.Furthermore, there was a
significant correlation between graphical representations and the students’
spatial performance.

Keller and Hirsch (1998) identified
several factors that influence the preference of representations.These factors included: (a) the nature of
students’ experiences with each representation, (b) the students’ perceptions
of the acceptability of using a representation, and (c) the level of the
task.Another theories concerning
representations’ preferences comes from the research done by Donnelly (1995),
Dufour-Janvier, et al. (1987), Eisenberg and Dreyfus (1991), Poppe (1993),
Porzio (1994), and Vinner (1989).Özgün-Koca (1998, p. 5) summarized the previous findings of research in
reasons for students’ preferences for representations.These reasons were classified in two
sections: internal and external effects.In the first sections are: personal preferences, previous experience,
previous knowledge, beliefs about mathematics, and rote learning.Under external effects there are:
presentation of problem, problem itself, sequential mathematics curriculum,
dominance of algebraic representation in teaching, and technology and graphing
utilities.

Connections Among Representations

An issue widely discussed in the
consulted literature has been the connections between representations.Other authors have referred to it as translations
and linking processes among representations.

Dufour-Janvier and colleagues (1987)
realized that often students confront problems to see the same task when
different representations of the same problem are given.Students think that there are equal numbers
of problems as there are representations.These researchers presented the following situation:

A child resolves a problem using a
representation.We then show him to the
same problem resolved by someone else using a different representation.When we deliberately show him the answer of
this other child (the answer happens to be incorrect); a number of children are
not at all disturbed and find this quite natural because in their view the
first problem was done one way and the second done in another way. (p. 114)

With this
example it has been shown “that students do not see all of the representations
accompanying a single task as different ways for tackling the same situation”
(Porzio, 1994, p. 45).The literature
shows that students are able to work with different types of
representations.The troubles start
when they try to relate similar information provided by different
representations.Lesh, Post, and Behr
(1987) have stated that the connections between representations are
particularly important in order to solve problems.

Porzio (1994) conducted research
exploring the students’ abilities to see or make connections between graphical,
numerical, and symbolic representations in the context of problem situations,
using three different approaches: (a) traditional approach, (b) graphic
calculator approach, and (c) Calculus & Mathematica software.He found that in the traditional course
where symbolic representations were emphasized, students belonging to this
group exhibited the most difficulty of all the students in recognizing
connections between different representations and different forms of the same
non-symbolic representation.In the
group where graphics calculators were used and graphical and symbolic
representations were emphasized, students seemed to consider the main emphasis
to be on graphical representations.Finally, students who used the Calculus & Mathematica software,
where multiple representations which were illustrated in the majority of the
times as symbolic and graphic, were better than the other students at
recognizing connections between different representations and varied forms of
the same representation.Also, students
often used graphical/symbolic and symbolic/numerical representations.The research results from Porzio can be
summarized as follow:

Students are able better able to see, or make, a
connection between different representations when one or more of the
representations is emphasized in the instructional approach that they
experienced and [underlined by the author] when then instructional
approaches includes having students solve problems specifically designed to
explore or establish the connection(s) between the representations. (p. 443)

Kaput (1989a), one of the recognized
researchers in the field has introduced the concept of linked
representations [italics added].He
describes the cognitive potential of dynamic links between
representations.Kaput said: “multiple,
linked representations of mathematical ideas likewise provide a form of
redundancy, a redundancy that can be exploited directly in a multiple, linked
representation learning environment” (p. 179).According to him, one of the advantages of using linked representations
is that they enable students to repress some aspects of complex ideas and give
more attention to others, supporting the varied ways of the learning and
reasoning process.

Janvier and colleagues (1993) have
introduced the term translation in the discussion of representation in
mathematics.They argue that the
process of translation from one representation to another is possible as the
result of the synonymy phenomenon presented earlier.These researchers think that in order to teach the translations
skills efficiently it is necessary that students view the translations from
both directions.Janvier, et al.
suggests, for example, that opposite translations, that is “graph à
formula” and “formula à graph” should be tackled in pairs (p. 98).

Hiebert and Carpenter (1992) have
conducted extensive research dealing with teaching and learning mathematics
with understanding.They have devoted
some sections in their research to connections between representations.They argued that connections between external
representations of mathematical concepts could be constructed by the student “between
different representations forms of the same mathematical idea or between
related ideas within the same representation form” (p. 66).Hiebert and Carpenter said that the
connections between different representations are possible if they are based on
the relationships of similarity (“these are alike in the following ways”) and
in the relationships of difference (“these are different in the following
ways”) (p. 66).The particular
connections between representations can be constructed, according to these
researchers, looking carefully at how they are the same and how they are
different.Finally, Hiebert and
Carpenter affirm that the process of connections between representations plays
a particular role in learning mathematics with understanding.

Representations and Understanding

Understanding and meaning are two
key terms in mathematics teaching and learning.They have been reinforced in the current reform movements.On this topic, Goldin and Shteingold (2001)
affirm “conceptual understanding consists in the power and flexibility of the
internal representations, including the richness of the relationships among
different kinds of representation” (p. 8).Janvier et al. (1993) mentions that in any discussion about theories of
representation, two terms are transcendental: “to mean” and “to
signify” [italics added] (p. 81).

Porzio (1994) points out that the
theoretical framework that support the use of multiple representations in
mathematics comes from the research done by Hiebert and Carpenter (1992).These researchers affirm that understanding
can be described in terms of internal knowledge structures.They define understanding in mathematics as
follows:

A mathematical idea or procedure or fact is
understood if it is part of an internal network.More specifically, the mathematics is understood if its mental
representation is part of a network of representations.The degree of understanding is determined by
the number and strength of the connections.A mathematical idea, or procedure, or fact is understood thoroughly if
it is linked to existing networks with stronger and numerous connections.(Hiebert and Carpenter, 1992, p. 67)

Based on
this definition of understanding, Porzio says that one of the principal goals
of mathematics teaching and learning is to provide tools and opportunities to
students in order that they can develop large and well-connected internal
networks of representations.

Goldin and Shteingold (2001) remark
that:

A mathematical representation cannot be understood
in isolation. A specific formula or
equation, a concrete arrangement of base-ten blocks, or a particular graph in
Cartesian coordinates makes sense only as part of a wider system
[italics added by the author] within which meanings and conventions have been
established.The representational
systems important to mathematics and its learning have structure, so
that different representations within a system are richly related to one
another. (p. 2)

Kaput (1989b) describes as
epistemological sources of mathematical meaning the connections that could be
possible between representations.He
identifies the following factors as the epistemological sources of mathematical
meaning:

1.transformations
within and operations on a particular representation system;

2.translations
across mathematical representation systems;

3.translations
between non-mathematically described situations and mathematical representation
systems; and

4.consolidation
and reification of actions, procedures, or webs of related concepts into
phenomenological objects that can then serve as the bases of new actions,
procedures, and concepts at a higher level of organization. (p. 106)

Porzio
(1994) points out that the first three sources of mathematical meaning
identified by Kaput correspond to the many kinds of connections that can be
made between distinct forms of the same type of representation and between
different kinds of representations.

Interpretation of Representations

The final topic concerning multiple
representations deals with the interpretation of these representations in
mathematics.This topic is one of the
most widely discussed.As cited in the
work by Janvier and colleagues (1993, p. 81), Von Glasersfeld (1987, p. 216)
affirms: “A representation does not represent by itself – it needs
interpreting; to be interpreted, it needs an interpreter”.Greeno and Hall (1997) mention that in order
to interpret representations, students should be involved in a learning
environment where complex practices of communication and reasoning are
emphasized.

The literature agrees in finding
that graphs, tables, pictures, and diagrams, among others, do not constitute a
representation by themselves.Greeno
and Hall (1997) citing the research done decades ago by Charles Sanders Peirce
(1955) said, “for a notation to function as representation, someone has to
interpret it and thereby give it meaning” (p. 366).Peirce identified three factors involved in representation: (a)
something that is represented, the referent; (b) the referring expression that
represents the referent; and (c) the interpretation that links the referring
expression to the referent.Following
Peirce’s principle, Greeno and Hall say that notations such as tables,
equations, and graphs are considered as potential representations.They become representations per se when
someone gives them meaning by interpreting them.

Greeno and Hall consider equations,
Cartesian graphs, and tables as standard forms of representations and they have
frequently shared conventions of interpretation.These researchers indicate that the process of learning these
conventions are important for students in order to encounter, construct and
communicate their ideas.

Standard instructional practices in mathematics
provide students with opportunities to learn the conventions of interpretation
of standard representational forms at an operational level.Teachers explain how to construct and
interpret tables, graphs, and equations, and students are asked to construct
representations of given information in these forms and to interpret representations
that they are given.In these
activities students can learn to follow the standards conventions of
interpretation for the forms, and with this learning the forms function as
representations for the students. (p. 366)

According
to these researchers, a practice like this one is now promoting the recognition
of interpretation as an essential part of representations in mathematics.These activities serve to give students the
opportunity to learn how to follow standard conventions of interpretation, and
moreover, how to understand how representations work.

As stated earlier, the use of
multiple representations in mathematics is strongly linked to the learning of
important mathematics concepts.This
section will describe some research of theorists and their contributions to
this field.

One of the most recognized
researchers in this field is Zoltan P. Dienes.His extensive work in theories of learning has impacted mathematics
teaching and many of his ideas are still been applied today in educational
settings (English and Halford, 1995).As cited in Gningue (2000, p. 59), “Dienes (1971) believes that
abstraction results from the passage of concrete manipulations of objects to
representational mapping of such manipulations and then to formalizing such
representations into rule structures”.Based on this belief, Dienes elaborated his four general principles for
teaching concepts.

The two first Dienes’ principles are
the Dynamic Principle and the Constructivity Principle.He thinks that the best way to teach a new
concept is through the formulation of a particular situation where students are
lead to constructive, rather than analytical thinking and understanding
(Gningue, 2000).The third principle is
the Mathematical Variability Principle.It states “concepts involving variables should be learned by experiences
involving variables should be learned by experiences involving the largest
possible number of variables” (Dienes, 1971, p. 31).Lastly, the Perceptual Variability Principle or Multiple
Embodiment Principle “demands a richness of concrete experiences with the same
conceptual structure, so that children may glean the essentially abstract
mathematical idea, which must be learned.To allow as much scope as possible for individual variations in
concept-formation, as well as to induce children to gather the mathematical
essence of abstraction, the same conceptual structure should be presented in
the form of as many perceptual equivalents as possible” (pp. 30-31).

This principle suggests that the learning of a mathematical concept
reaches its maximum expression when students are exposed to a concept using a
variety of physical materials or embodiments [italics added].Resnick and Ford (1981) said: “multiple
embodiments are viewed as facilitating the sorting and classifying process that
constitutes the abstraction of a concept.Seeing a principle operating similarly even when different materials are
used seems to help children discover what is and is not relevant to the
concept” (p. 121).These researchers
point out that the students’ familiarity with the various mathematical
materials is an assumption of presenting concepts using multiple
embodiments.Resnick and Ford argue
that if this familiarity process does not occur first, the use of embodiments
will be “counterproductive” (p. 121) since students should learn the materials
and a new mathematical principle at the same time.According to Dienes, as cited in Resnick and Ford’s research,
multiple embodiments should look different from each other in order that
children can observe the structure from many different perspectives and
construct a vast amount of mental images about each concept.The use of these embodiments should allow
manipulation of all variables related with the concept under study.

Dienes (1973) clarified his four principles by pointing out six stages of
teaching and learning mathematical concepts.Similar to the intellectual developmental stages introduced by Piaget,
Dienes affirmed that the learning of mathematical concepts occur through
sequential stages.These stages are:
(a) free play, (b) games, (c) searching for communalities, (d) representation,
(e) symbolization, and (f) formalization.As mentioned in Gningue (2000), the first three stages are described as
components of the first Dienes’ principle.The second phase of the learning cycle promoted by Dienes constitutes
the transition process from manipulative materials to abstract representations.These representations are illustrated
initially as pictorial models and graphs, and finally as mathematical
symbols.The beginning of this second
phase is the fourth or representation stage.

The child needs to develop, or to receive from teacher,
a single representation of the concept that embodies all the common elements
found in each example.This could be a
diagrammatic representation of the concept, a verbal representation, or an
inclusive example.Students need a
representation in order to sort out the common elements present in all examples
of the concept.A representation of the
concept will be usually more abstract than the examples will bring students
closer to understanding the abstract mathematical structure underlying the concept.(Gningue, 2000, p. 64)

The fifth
stage described by Dienes is where the students describe the representation of
the concept verbally and using mathematical symbols.Dienes suggests that the teacher should supervise the use and
construction of symbols.Students can
use their own symbols, but they should be aligned with those included in the
textbook.

Janvier and colleagues (1993) affirm that students do not always
appreciate and accept that two or more external representations belong to the
same concept.Rather, students have
exhibited the preference to work mainly “on a one-to-one correspondence basis”
(p. 91).Janvier et al. mention that
opponents of Dienes’ principles state that adding more embodiments to concept
instruction is not a guarantee that students will get a better and more
meaningful internal representation of the concept.

Constructivism has had an enormous impact on current education learning
theories, and mathematics instruction is no exception.De Jong and colleagues (1998) said “modern
education learners are encouraged to construct their own knowledge, instead of
copying it from an authority, be it a book or a teacher” (p. 9).Hart (1991) mentions “constructivist theory
suggests that knowledge is actively constructed out of one’s experiences” (p.
4).Noddings (1990) explains that
constructivism has basically two main characteristics: (a) a cognitive
position, and (b) a methodological perspective.This review will focus on the first characteristic of
constructivism.She affirms: “as a
cognitive position, constructivism holds that all knowledge is constructed and
that the instruments of construction include cognitive structures that are
either innate or are themselves products of developmental construction” (p. 7).

Noddings (1990) in her extensive work in the field, has summarized in the
following points the current constructivists views:

1.All
knowledge is constructed.Mathematical
knowledge is constructed, at least in part, through a process of reflective
abstraction.

2.There
exist cognitive structures that are activated in the processes of construction.

Technology has the potential to
completely change current trends in teaching and learning of mathematics.Researchers as De Jong and colleagues (1998)
have agreed with the need for technology in mathematics scenarios.They said:“Technology plays a major role in implementing these new trends in
education” (p. 9).As cited in Gningue
(2000), Fey (1989) proposed the use of a vast amount of technological
resources, such as calculators, computers, and computer software to teach
concepts in algebra.According to him,
“the most obvious implication of computer tool software is the opportunity to
rebalance the relationship among skill, understanding, and problem-solving
objectives in algebra” (p. 204-205).Findings from research studies conducted by Orton (1983a, 1983b) and
Tall (1985) indicate that the use of technology is advantageous in order to
promote conceptual understanding.

One of the main advantages to the
uses of technology in mathematics education is, without a doubt, the capability
to present information in multiple representations.Mathematical concepts can be introduced through the use of
tables, graphs, equations, and other representations.Keller and Hirsch (1998) affirm that the incorporation of
multiple representations supported by technology is an important topic in
mathematics curricula.Important lines
of research conducted by recognized scholars such as Fey (1989), Goldenberg
(1987), Kaput (1992), and Porzio (1994) indicate that the access to multiple
representations of mathematical concepts has increased with the advancements of
technology.

Use of technology in the classrooms appears to
affect student learning in a positive way.Those students using technology to access multiple representations may
have “richer” concept images than those who do not have the same experience…

Technology can provide a means for presenting
concepts via multiple representations and for students to work within multiple
representations.A review of the
literature indicated there may be some positive effects from the use of
technology, capable of graphing and/or symbolic manipulation, in the classroom.
(Hart, 1991, pp. 45-46)

Several mathematics reform projects have
been developed nationwide in order to promote teaching mathematical concepts
using multiple representations supported by technology.One of them is the Harvard Calculus Project,
also called “Rule of Three”, which emphasizes the use of three representations:
graphical, numerical, and symbolic (Hart, 1991; Porzio, 1994).Hughes-Hallet (1991) indicated:

[The philosophy of the project] is based on the
belief that in order to understand an idea, students need to see it from
several points of view, and to build web connections between the different
viewpoints.I believe that in calculus
most of the ideas should be presented in three ways: graphically and
numerically, as well as in the traditional algebraic way.Technology is invaluable here. (p. 33)

Porzio
(1994) points out that there are differences between students participating in
curriculum projects, specifically in calculus (Tucker, 1990), where they are
using computers, calculators and representations where graphics and symbols are
also emphasized, students using the Calculus & Mathematica software where
technology is used intensively, and students from traditional approaches.Nevertheless, he states that there is little
evidence of the effectiveness of these technological approaches.

Fey (1989) affirms that the use of
numerical, graphic and symbol manipulation is a powerful technique for
mathematics teaching and learning.He
identified several ways in which computer-based representations of mathematical
ideas are unique tools for problem solving.These are:

1.Computer
representations of mathematical ideas and procedures can be made dynamic in
ways that no text or chalkboard diagram can.

2.The
computer makes it possible to offer individual students a work environment with
representations that are flexible, but at the same time constrained to give
corrective feedback to each individual user whenever appropriate.

3.While
some multiple embodiment computer programs might be viewed as poor simulations
of more appropriate tactile activity, it has been suggested that this
electronic representation plays a role in helping move students from concrete
thinking about an idea or procedure to an ultimately more powerful abstract
symbolic form.

4.The
versatility of computer graphics has made it possible to give entirely new
kinds of representations for mathematics.

The concept of function is one of
the key topics in mathematics.It
dominates the mathematical panorama and is present in a vast part of the
instructional activities developed at secondary and college levels.Thorpe (1989) proposes the study and the use
of functions as “the centerpiece of algebra instruction because functions are
at the very heart of calculus” (p. 11).Selden and Selden (1992) coincide with Thorpe, when they say that functions
play a central and unifying role in mathematics.As cited in Hart (1991, p. 10), Vinner and Dreyfus (1989)
introduced the Dirichlet-Bourbaki definition of what a functions is.It says: “a function is a correspondence
between two nonempty sets that assigns to every element in the first set (the
domain) exactly one element in the second set (the codomain)” (p. 357).This definition has been kept and taught in
the majority of the mathematics curricula (Lloyd and Wilson, 1998).

Further, the concept of function has
the capability of being taught using different representations.The literature illustrates functions in
several ways, such as mapping diagrams, tables, graphs, and equations.All of these representations are primarily
intended to promote a better understanding of the concept.Research done by Sfard (1987) indicates that
in order to get a good concept of functions, students should develop an
operational before a structural concept.After this, students will benefit from the introduction of functions
using the different representations, such as mappings, tables, and graphs
(Poppe, 1993, p. 26).

According to Poppe (1993), tables,
graphs, and mapping diagrams are representations of functions that can be used
to create mental structures.

The computational processes of creating tables,
graphs, and mapping diagrams would afford the students an opportunity to
develop an operational conception of function.The exploration of the function idea in a concrete context using tables,
graphs, and mapping diagrams provides the students with a richer foundation for
development of the variable concept. (p. 25)

Thomas (1975) examined the aspect of
understanding of functions in students from seventh and eighth grades,
identifying five stages in the development of the concept of function:

3.Operational
ability in finding images, pre-images, range, and domains where the mappings
are given by some display of the set of ordered pairs.

4.Identification
of noninstances of mappings with finite domains.

5.Composition
of mappings and the translation from one representation of mapping to another.
(Poppe, 1993, p. 21)

Markovits, Eylon, and Bruckheimer (1986) found that most students understood
that a function would have more than one representation.They stated that almost fifty percent of
their study population was able to identify two functions, one in algebraic
form and the other in graphical form, as being the same.In addition, several studies have been done
comparing the difference between the uses of two or more representations of
functions.Iannone (1975) compared
tabular approach and mapping diagrams of functions.Results show that the best way to represent the function concept
is through the use of mapping diagrams.Poppe (1993) conducted research in this specific area and found that
students were aware of the uses of tables, graphs, and mapping diagrams, and
tables were helpful in finding generalized patterns.On the other hand, students found tables, graphs, and mapping
diagrams helpful. In conclusion, the use of tables, graphs, and mapping
diagrams aided instruction.Students
had the opportunity to see the same information in different ways.

Results from Markovits et al. (1986) also show that difficulties arose
when students managed more than one representation of functions at the same
time.They pointed out, for example,
that students changed domain and codomain of some functions.Goldenberg (1988) affirms that confusion may
occur trying to relate information provided by two different
representations.He suggests an
appropriate transfer process between the representations.Hart (1991) introduced the term compartmentalization
[italics added] when students do not relate several representations for the
same function.A lack of connections
between two representations –graphical and algebraic– was found in research
conducted by Dreyfus and Eisenberg (1988).Ferrini-Mundy and Graham (1991) found similar results when students
managed algebraic and graphical contexts as separate worlds.Recognizing troubles shown by students
trying to relate representations of the same function, Poppe (1993) affirmed
that: “students needed more opportunity working with the different representations”
(p. 98).

The previous sections have described
research studies and current trends on the uses of multiple representations in
mathematics teaching and learning.Preferences, connections, among others, were also discussed.Theories of learning that support the use of
representations in mathematics were introduced and discussed.Further, research studies dealing with how
the available technologies have been used to promote understanding through representations
in mathematics were included in this chapter.Finally, the importance of functions in the curricula and a view of
their representations were discussed.

The next chapter will present the methodology of this research project,
including participants, settings, and instruments used to obtain data.The procedures followed in the instructional
activities will also be described, as well as the statistical tests used to
answer the questions of this investigation.